DescriptionProving the Poincare conjecture was worth $1 million and a Fields medal to a man who turned down both.The Poincare conjecture roughly states that if you have a 3-dimensional space M in which every loop can be shrunk to a point, then M must be a 3-sphere. But what is meant here by a 3-dimensional space? Are there in fact any 3-dimensional spaces in which you cannot shrink all loops to a point? Why should we anticipate that the conjecture is true? In contrast to 2-dimensional spaces, 3-dimensional spaces can differ amongst each other in subtle ways. We will take the Poincare conjecture as a motivation to try to understand, describe, distinguish, and catalogue 3-dimensional spaces using various techniques. Our methods will most often be highly visual - cutting up and gluing together pieces that we understand to make new spaces with interesting properties.
Co-requisitesGeometric Topology III.ResourcesYou can look up the wikipedia articles on Dehn surgery and Heegaard splitting to get some idea of the techniques used in constructing 3-dimensional spaces.Relevant papers and books include the following:
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email: Andrew Lobb